It's hard to overestimate the influence Alcuin of York (c. 732-804) had on Western civilization. He also left the earliest known European collection of puzzles, Propositiones alcuini doctoris caroli magni imperatoris ad acuendos juvenes - Propositions by Alcuin Teacher to the Great Emperor Charles to Sharpen up the Young. The collection consists of 53 problems (some of which became classic, like the problem #18 of moving Cabage, Goat, and Wolf to the other shore of a river constrained by a small size boat and unfriendly attitude of the actors toward each other.)

Here I am concerned with problem #12:

A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil; another 10 were half full, while another 10 were empty. Divide the oil and flasks so that an equal share of the commodities should equally come down to the three sons, both of oil and glass.

To each son will come ten flasks as his portion. But divided them as follows; give the first son the ten half-full flasks; then to the second give five full and five empty flasks, and similarly to the third.

Martin Erickson points out that this is not the only possible solution and asks how many there are. There in fact five distinct solutions, not counting the permutations as distinct. Can you find them?

At a recent dinner I posed this problem to my 8-grader boy. With a twinkle in his eye he immediately suggested to fill five empty flasks with the contents of the ten half-empty ones, making the solution obvious and, in some sense, fairer than other solutions. Not only each of the sons receives equal number of bottles and equal amounts of oil - they receive them in identical packages.

The boy knew that his solution was not what was expected but enjoyed himself on my account. Right away, he proceeded with another solution that was that of Alcuin's. Then, with a little prompting, he came up with the remaining solutions.

You know, I was happy he did not ask that pernicious question, What is it good for? We may speculate that the question occurred to neither Alcuin himself nor to Charlemagne (to whom the problems have been sent occasionally, one at a time, and later collected in a book). After all, Charlemagne made Alcuin his effective Secretary of Education, even though some of the problems have been plain jokes. E.g., #14 reads

An ox ploughs a field all day. How many footprints does he leave in the last furrow.

The ox leaves no trace in the last furrow, because he precedes the plough. However many footprints he makes in the earth as he goes forward, the cultivating plough destroys them all as it follows. Thus no footprint is revealed in the last furrow.

Another one (#43) does not require an answer:

A certain man had 300 pigs. He ordered all of them slaughtered in three days, but with an uneven number being killed each day. He wished the same thing to be done with 30 pigs. What odd number of pigs out of 300 or 30 were to be killed in three days? (This ratio is indissoluble and was composed for rebuking.)

Alcuin is not remembered because of his problem book but, ironically, another great - Leonardo of Pisa - who lived 400 years later, has his name immortalized in the connection with a certain puzzle, not the accounting revolution his work engendered. (See, Devlin.)

Nonetheless, in time, some of Alcuin's problems got involved with a good deal of mathematics. The flasks inheritance problem has been extended to a generic "n full, n half-empty, and n empty flasks." The number of solution to each is usually denoted t(n) and the sequence {t(n)} is known as Alcuin's sequence. The sequence satisfies an 8-term recurrence relation and has a nice generating function [Erickson, p. 91]. It comes up as a number of integer triangles with a given perimeter. In [Yaglom & Yaglom, #30], that latter problem is marked with two stars as very difficult.

Many of Alcuin's problems were too simple to leave any trace in history; most were rather artificial; it is not known whether any were original. This is inspiring, however, to have access to this collection, as it throws - if only a quantum of - light on the life of this great man. Looks like he found amusement in mathematics.

Solition

A solution can be expressed by the number of full flasks received by each son. For example, Alcuin's solution could be expressed as {0, 5, 5}. A triple {1, 4, 5} tells us that the first son receives 1 full flasks, and to make up for the required quantities, 8 half-full and 1 empty flask are added. The second son gets 4 full, 2 half-empty and 4 empty flasks, the last one gets 5 of each full and empty flasks. The other solutions in that form are {2, 3, 5}, {2, 4, 4}, and {3, 3, 4}.